Metamaterials are attracting increasing attention due to their ability to support novel and engineerable electromagnetic functionalities. In this paper, we investigate one of these functionalities, i.e. the extraordinary optical transmittance (EOT) effect based on silicon nitride (Si3N4) membranes patterned with a periodic lattice of micrometric holes. Here, the coupling between the incoming electromagnetic wave and a Si3N4 optical phonon located around 900 cm-1 triggers an increase of the transmitted infrared intensity in an otherwise opaque spectral region. Different hole sizes are investigated suggesting that the mediating mechanism responsible for this phenomenon is the excitation of a phonon-polariton mode. The electric field distribution around the holes is further investigated by numerical simulations and nano-IR measurements based on a Scattering-Scanning Near Field Microscope (s-SNOM) technique, confirming the phonon-polariton origin of the EOT effect. Being membrane technologies at the core of a broad range of applications, the confinement of IR radiation at the membrane surface provides this technology platform with a novel light-matter interaction functionality.